Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 2G / a / Solution

The sequence has uniform convergence on $X$ if there is a function $f:X\to \mathbb{R}$ such thatThe same $N$ must work for every $x$. Equivalently, the supremum of the error over $X$ tends to zero. In pointwise convergence, $N$ is allowed to depend on $x$.